Fractal Space-Time from Spin-Foams
In this paper we perform the calculation of the spectral dimension of spacetime in 4d quantum gravity using the Barrett-Crane (BC) spinfoam model. We realize this considering a very simple decomposition of the 4d spacetime already used in the graviton propagator calculation and we introduce a boundary state which selects a classical geometry on the boundary. We obtain that the spectral dimension of the spacetime runs from to when the energy of a probe scalar field decreases from high to low energy. The spectral dimension at the Planck scale depends on the areas spectrum used in the calculation. For three different spectra , and we find respectively dimension , and .
In past years many approaches to quantum gravity studied the fractal properties of quantum spacetime. In particular in causal dynamical triangulation (CDT) cdt () and asymptotically safe quantum gravity (ASQG) asqg (), a fractal analysis of spacetime gives a two dimensional effective manifold at high energy. In both approaches the spectral dimension is at small scales and at large scales. The previous ideas have been applied also in the context of noncommutative geometry to a quantum sphere and -Minkowski dario (), in causal sets DR1 ()DR2 () and in loop quantum gravity Modesto:2008jz (). See Carlip:2009kf () for a summary and new insights. Spectral properties have been considered also for the cosmology of a Lifshitz universe calcagni (). Spectral analysis is a useful tool to understand the effective form of space at small and large scales. We believe that the fractal analysis could be also a useful tool to predict the behaviour of -point correlation functions at small scales propagatoreR ()propagatoreMR () and to attack the singularity problems of general relativity in the full theory of quantum gravity CBH-AB ()CBH-M ().
In this paper we apply to the Barrett-Crane (BC) spin foam model barrett () for Riemannian 4d quantum gravity the analysis introduced in Modesto:2008jz (). The fractal properties of 3d spacetime were studied in fractal3d (). The main ingredient is the general boundary formalism which provides quantum amplitudes associated to a finite spacetime region for a given boundary 3-geometry gb1 ()gb2 (). The formalism is implemented in the same spirit of the calculation of LQG graviton propagator propagatoreMR ()propagatoreR ()propagatoreBMRS ()propagatoreBMP ()propagatore3d ()propagatoreAR1 ()propagatoreAR2 ()propagatoreABR3 (). For our purposes we will consider a boundary state peaked over the boundary geometry of a single -simplex.
The paper is organized as follows. In the first section we define the framework and recall the definition of spectral dimension in diffusion processes. In the second section we define the spectral dimension in quantum gravity. The analysis is general and not strongly related to the specific models. We continue calculating explicitly the spectral dimension for the BC theory using the general boundary formalism to define the 4d quantum gravity path integral.
Spectral dimension in diffusion processes.
The following definition of fractal dimension is borrowed from the theory of diffusion processes on fractals avra () and easily adapted to the quantum gravity context. Let us study the Brownian motion of a test particle moving on a -dimensional Riemannian manifold with a fixed smooth metric . The probability density for the particle to diffuse from to during the fictitious time (this is just a fictitious time since we are probing the spacetime properties, not only the properties of space) is the heat-kernel , which satisfies the heat equation
where denotes the covariant Laplacian:
The heat-kernel is a matrix element of the operator , acting on the real Hilbert space , between position eigenstates
Its trace per unit volume,
has the interpretation of an average return probability. Here denotes the total volume. It is well known that possesses an asymptotic expansion for of the form . The coefficients have a geometric meaning, i.e. is the volume of the manifold and if then is proportional to the Euler characteristic. From the knowledge of the function one can recover the dimensionality of the manifold as the limit for small of
If we consider arbitrary fictitious times , this quantity may depend on the scale we are probing. (5) is the definition of fractal dimension we will use.
Spectral dimension in quantum gravity.
In quantum gravity it is natural to replace the Laplacian in the heat-kernel equation (1) with an effective Laplacian associated to the expectation value of the operator . It is defined by the path integral
Including quantum gravity effects.
We expect the effective Laplacian to run with the probed energy scale when quantum gravity effects become important. Above some energy scale the quantum spacetime reveals its fuziness. So we expect that
where is the Laplacian at a reference infrared momentum , and is a scaling function of the energy scale. For instance, in the case the i.r. Laplacian is the flat one, we have, expading in Fourier modes:
We argue that the scales and should be physically identified, since represents the energy probed by the th mode of a scalar field. Then we define an average return probability which includes quantum gravity effects by
If, as we will find, , then
and the fractal dimension is .
Spectral dimension in spin foams.
In the context of spin foam models for quantum gravity we can give a precise meaning to the formal path integral (6) and define the spectral dimension in the general boundary formalism. We introduce a gaussian state peaked on the boundary geometry defined by the 3-metric and the conjugate momentum. We can think the boundary geometry to be the boundary of a -dimensional ball. The state is Gaussian and symbolically given by:
The amplitude (6) can be defined for a general spin foam model in the general boundary framework as
where codifies the spin foam dynamics. We will consider the BC model with vertex amplitude proportional to the -symbol .
The fundamental ingredient required to study the fractal properties of quantum spacetime is the the scaling of the metric. To this purpose we consider only one simplex that could be part of a more complicated simplicial decomposition. In other words we suppose spacetime to be a 3-ball with a boundary 3-sphere and triangulate the sphere in a very fine way (the dual of the triangulation is a spin-network). Then we consider another 3-ball and its boundary 3-sphere but at a smaller scale. Since we are considering the 3-sphere at two different scales all the representations labeling the boundary spin-network will be rescaled conformally. In our simplification the 3-ball is approximated by a single 4-simplex and the boundary 3-sphere by the ten faces of the simplex. Summarizing: in 4d gravity we approximate a -ball with a single simplex and the boundary sphere with the surface of the simplex given by ten triangles.
We use a conformal metric defined by , where is a background metric and is the conformal factor. This is the hypothesis made in Modesto:2008jz () and fractal3d (). In the quantum theory the conformal factor is replaced by the area operator and the metric operator is defined by . To extract the fractal properties of spacetime we approximate the metric in the Laplacian with the inverse of the area operator in 4d. In fact the Casimir operator is related to the area spectrum of a triangular face in the simplicial decomposition of space. We consider three possible area spectra
where is the representation dual to the triangular face. The quantity which is necessary to compute is
The boundary state in the notation above is
The 10 spins label the boundary triangles of a 4-simplex. The Gaussian is peaked over the homogeneous spin configuration, namely over an equilateral 4-simplex of size of order . The dihedral angles define the boundary extrinsic geometry. is a matrix with positive definite real part. We chose in the numerical calculation, but other choices do not affect qualitatively the analysis.
The expectation value (14) reads
where we introduced the following notation for the normalization, . In Fig.1 we have plotted the amplitude (18) as a function of the representation . This is also the quantum gravity scale defined by .
For the amplitude (18) is well approximated by the classical function
The result (19) can be obtained also analytically in the large limit replacing the amplitude with the exponential of the Regge action. More precisely, since the boundary state is peaked on large values of the spins, the -symbol in the sum (18) can be approximated with its large spin asymptotic formula
where is the Regge action for a 4-simplex. One of the two exponentials in which the cosine can be decomposed suppresses the sum (18) through the presence of a rapidly oscillating phase. The term in (20) is dominant but it is suppressed inside the sum (18) through a non trivial mechanism Bianchi:2007vf (). Expanding up to second order the result (19) is obtainded by a simple gaussian integration. We refer to the literature on the graviton propagator for this technique.
In the range we have interpolated the quantity (18) numerically. Data are fitted with the function , where , and ,
What we learnt from the explicit calculation of (18) can be summarized as follows,
where we introduced the infrared scale , and, by definition, . Actually (23) is correct also for the area spectra and but with different values of , and the proper functions and .
We introduce here a physical input to put the momentum in our analysis. If we want to observe the spacetime with a microscope of resolution (the infrared length is ) we must use a probing particle of momentum . The energy scaling property of the Laplacian, i.e. the scaling function , can be obtained by replacing: , and , where is an infrared energy cutoff and is the Planck energy. This gives
The term is added to facilitate the calculations of the spectral dimension. Notice that the scaling function represents also by construction the scaling of the inverse metric, .
The parameter introduced in the second section for the area operator is
The dependence from in (11) determines the fractal dimensionality of spacetime via (5). In the limits and where we are probing very large and small distances, respectively, we obtain the dimensionalities corresponding to the largest and smallest length scales possible. These limits are governed by the behaviour of for and , respectively.
The spectral dimension increases from to when the momentum of the probe field decreases from to . A plot of the fractal dimension is given in Fig.2. In the regime the spectral dimension is . We can repeat the analysis for the area spectra and . The spectral dimension is the same for or but we can have differences at the Planck scale. We summarize the result in the following table,
Correlations. The 2-point correlation function of the Brownian motion over a Riemannian -dimensional manifold is defined by
The heat kernel has an approximate expression for :
where is the geodesic distance and with “curv.” we have indicated the curvature corrections. For the result of the integration (26) is
We use the spectral dimension we found in the spin foam model as an effective dimension at high energies, hence in (28) we make the substitution . At low energy the spectral dimension is and . In the energy range where the spectral dimension is the propagator is: . At the Planck scale instead, defining ( for the three different area spectrum), .
Conclusions and Discussion. In this paper we computed the spectral dimension () of 4d quantum spacetime in the Barrett-Crane spinfoam model. We considered the simplest decomposition of spacetime and used the general boundary formalism to characterize the scaling properties of the expectation value for the traced propagation kernel. Our main observable is a conformal metric defined by , where is a background metric. In the quantum theory the conformal factor is quantized as a function of the area operator. We made our analysis with three kind of area spectra, obtaining different values for the spectral dimension at Planck scale (see table) but the same result for (see Fig.2).
We interpret the results in the following way. At high energy the spectral dimension is because the manifold presents holes typical of an atomic structure.
Acknowledgements. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation.
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